1 Fluid dynamic principles
1.3 Boundary layers, boundary layer control
C y p
x
A
Wx(y) W∞
B
x 0 w x 0 p
x <
∂
∂
∂ >
∂ x 0
w x 0 p
x>
∂
∂
∂ <
∂ xdx
p p
∂ +∂
x 0 w x 0 p
x=
∂
∂
∂ ≤
∂
3
4
5
6 2
τ0
W∞
1
Fig. 1.4. Boundary layers with positive and negative pressure gradients
too, becomes turbulent after a certain length of the flow path while a laminar sub- layer remains.
The transition from a laminar to a turbulent boundary layer (depicted in Fig. 1.5) depends on: Reynolds number, surface roughness, turbulence of the outer flow, pressure gradient and wall curvature.
Fig. 1.5. Boundary layer transition from laminar to turbulent
If the flow is accelerated (domain B in Fig. 1.4), wx/x > 0 and p/x < 0 ap- ply according to Eq. (1.7). The boundary layer thickness then decreases; in accel- erated flow correspondingly lower hydraulic losses are experienced. The maxi- mum shear stress on the wall occurs at p/x < 0; fuller velocity profiles are obtained with increasing acceleration. Accelerating inlet chambers therefore pro- duce more uniform inflows to the impeller of a pump.
In decelerated flow (Fig. 1.4, domain C) wx/x < 0 and accordingly p/x > 0 applies. The boundary layer thickness grows and the wall shear stress drops; it be- comes zero at a certain deceleration (profile 5 in Fig. 1.4). At this point the flow separates from the wall (it “stalls”), resulting in recirculation in the boundary layer further downstream (profile 6). Due to the separation the main flow contracts and accelerates in accordance with the reduced cross section. Downstream it inter- mixes with the wake through the exchange of momentum. These processes result
1.3 Boundary layers, boundary layer control 9
in correspondingly high hydraulic losses which generate a drop of the static pres- sure (refer to Chap. 1.6).
If the channel cross section is other than circular, the velocity distribution is not rotational-symmetric. Similar to Eq. (1.16), different velocity profiles are expected in sections A-A and B-B (diagonal) of a square channel (Fig. 1.6). Consequently, the wall shear stresses change over the channel perimeter. The shear stresses are lowest in the diagonal B-B and greatest in section A-A. Since the static pressure (with flow in straight channels) is constant over the cross section, the equation:
L 2 U c w
= L U
=
p τ Δ ρ f 2 Δ
Δ (U = wetted perimeter) (1.17)
can only be satisfied if fluid is transported through compensatory flows normal to the channel axis from locations with low shear stress to zones with higher shear stress. Such secondary flows occur, for example, in triangular or square channels (Fig. 1.6), hence also in the impeller or diffuser channels. Vortices (“corner vor- tices”) develop in the corners of a channel, between blades and shroud or between pillar and baseplate. Their effect can be observed when abrasive particles are car- ried with the flow giving rise to erosion, Chap. 14.5.
Corner vortices can also play a role in cavitation (Chap. 6), since the pressure in the vortex core drops relative to the pressure in the main flow, Chap. 1.4.2. The velocity components of the secondary flow attain approximately 1 to 2% of the velocity in mid channel, [1.2]. By rounding the corners the losses can be some- what reduced.
A - A B – B τa > τb
A
B
A
B
a b
w
Fig. 1.6. Secondary flow in a square channel
a) cross section, b) velocity distributions in sections A-A and B-B (diagonal)
There are a number of methods of boundary layer control that aim at reducing the flow resistance and retarding stall: (1) The point of separation in diffusers or on air foils can be shifted downstream through boundary layer extraction. (2) By injecting fluid, energy can be supplied to the boundary layer or separation en- forced. (3) Boundary layer fences may be designed to change the flow direction in the boundary layer in order to prevent the development of large vortices, or to sub- ject the boundary layer flow to the same deflection as the main flow. (4) By means of very fine longitudinal grooves it is possible to influence the vortex structure in
the boundary layer in a way that the flow resistance is reduced by several percent.
(5) The admixture of friction-reducing agents has a similar effect by reducing the thickness of the boundary layer. (6) Turbulence or swirl generators can be em- ployed to supply energy to the fluid near the wall through the exchange of mo- mentum. In this manner the boundary layer thickness, hence losses, as well as the separation tendency can be reduced.
The boundary layers in decelerated flow through pumps are mostly three- dimensional. In the impeller they are also subjected to centrifugal and Coriolis forces. Such complex conditions are not amenable to analytical treatment. How- ever, if an estimation of the boundary layer thickness is desired, the conditions on a flat plate without pressure gradients may be used for want of better information.
To this end the displacement thickness δ* may be used; it is defined by Eq. (1.18):
³δ
∞¸¸¹
¨¨ ·
©
§ −
≡ δ
0
x dy w 1 w
* (1.18)
The integration is performed up to the boundary layer thickness δ which is defined by the point where the local velocity reaches 99% of the mean velocity of the core flow. The displacement thickness with laminar or turbulent flow over a hydrauli- cally smooth plate is obtained from:
lam
x
1.72 x
* =
δ Re turb 0.139
x
0.0174 x
* =
δ Re x x
w x with: Re =
ν (1.19a, b, c) Here x is the length of the flow path calculated from the leading edge of the plate or component. Depending on the turbulence of the upstream flow and the rough- ness of the plate, the change from a laminar to a turbulent boundary layer occurs in the range of 2×104 < Rex < 2×106, refer also to Eq. (1.33b). These equations show that the boundary layer thickness grows from the leading edge at x = 0 with the length of the flow path and the viscosity of the fluid. The above formulae can also be used for the inlet into a channel or other components, provided the bound- ary layer thickness is significantly less than half the hydraulic diameter of the channel so that the boundary layers of opposite walls do not exert a noticeable ef- fect on each other. Consider for example an impeller with an outlet width of b2 = 20 mm and a blade length of L = x = 250 mm delivering cold water at a rela- tive velocity of wx =24 m/s. From Eqs. (1.19b and c) we find at the impeller out- let: Rex = 6×106 and į*turb = 0.5 mm. Since δ* is much smaller than b2 no appre- ciable interaction of rear and front shroud boundary layers would be expected.
Boundary layer considerations based on Eqs. (1.19a to 1.19c) would therefore be relevant as a first approximation.
The inlet length Le necessary to establish fully developed flow in pipes can be determined from Eq. (1.19d), [1.5] (in laminar flow Le is much longer than in tur- bulent flow):
e h
L 14.2 log Re 46
D = − valid forRe > 104 with c Dh Re =
ν (1.19d)